Principal specializations of Schubert polynomials, multi-layered permutations and asymptotics

Let $v(n)$ be the largest principal specialization of Schubert polynomials for layered permutations $v(n):={\max }_{w\in {\mathcal{L}}_n}{\mathfrak{S}}_w(1,\dots ,1)$. Morales, Pak and Panova proved that there is a limit $$\underset{n\to \mathrm{\infty }}{\lim }\frac{\log v(n)}{n^2},$$ and gave a precise description of layered permutations reaching the maximum. In this paper, we extend Morales Pak and Panova’s results to generalized principal specialization ${\mathfrak{S}}_w(1,q,q^2,\dots )$ for multi-layered permutations when $q$ equals a root of unity.